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OCR for page 333
24TH Symposium on Naval Hydrodynamics
Fukuoka, JAPAN, 8-13 July 2002
Whipping loads due to aft body slamming
G.K. Kapsenbergl, A.P. van 't Veerl, J.P. Hackett
and M.M.D. Levadou1
(1 Maritime Research Institute Netheriands,
2 Northrop Grumman Ship Systems)
ABSTRACT
This paper describes a method to measure
slamming loads on the aft body of a ship model in
the towing tank. The method uses a large number
of pressure gauges in the flat part of the stern. The
impact force is derived from the pressures by
integration. Experiments have been carried out on a
scaled model of a modern cruise vessel. The model
consisted of two segments so that a simplified 2-
node vibration mode is simulated. A structural
mathematical model was made of the model as
used for the tests.
The analysis shows that the measured
pressures can be used to predict the whipping loads
on the segmented model. It is concluded that the
measured loads can also be used on a finite element
model of the ships construction to predict whipping
stresses. Whipping stresses are important for
fatigue loads and the extreme vertical hull bending
moment.
INTRODUCTION
The goal of every commercial ship owner is to
maximize the revenue producing portion of the
vessel and minimize the non-revenue producing
portion such as the engine room. A "shoe box" with
no propulsion plant, ballast tanks, etc. would be
considered the ideal solution. As a result, over the
years an increasing demand was placed on design
naval architects to produce ship designs with full
beam main decks along almost the entire length of
the ship. For containerships this allows more
containers to be stowed on main deck. Likewise,
for cruise ships, a full width main deck allows a
full width superstructure that contains more
passenger cabins. Competition is now so intense
that this trend has extended to include filling out
the aft portions of the hull above the full load
waterline and below main deck. This added hull
volume means more container stowage capacity for
container ships, more space for passengers on
cruise ships, and a full breadth vehicle deck for
more vehicle stowage on ferries. Owners
rationalize that as long as the filling out of the hull
occurs above the full load waterline, there would be
no negative impact on calm water powering and
fuel economy. This design philosophy has
produced hull forms with progressively flatter and
flatter sterns. Some ships have been built with zero
or near zero deadrise angles and buttock angles
near the transom. Such shapes have hydrodynamic
benefits in calm water as demonstrated by
Hamalainen and van Heerd (1998~; however, they
are a far cry from the traditional cruiser sterns seen
on the Mariner hull form.
At the same time that this hull form
development was occurring, a new propulsion
system was being developed. The idea was to move
the electric propulsion motor from inside the hull to
an azimuthing pod located external to the hull.
This allows the elimination of the propeller shaft
and shaft struts, the rudder, and stern tunnel
thrusters. This concept provides greater flexibility
for the design engineer to locate elements of the
propulsion system; hence, an ability to provide
additional internal volume to the revenue producing
portion of the vessel. It also produces 10 to 15%
improvement in calm water powering and
significant improvements in vessel harbor
maneuverability.
The first ship equipped with podded
propulsion was the 'Seili', an ice going supply
vessel built by Kv~erner Masa in 1991; the first
application to a cruise vessel was in 1997, the
'Elation'. Since then podded propulsion has
become the industry standard on cruise ships. The
combination of the pod with the flat stern has
resulted in gains in propulsion efficiency. The
azimuthing pods encouraged design naval
architects to create stern designs that were even
OCR for page 334
flatter than their conventional screw propelled
sister ships to simplify the pod installation.
THE SLAMMING PROBLEM
It was not long before a serious drawback of
the flatter stern design surfaced. Some of the new
cruise ship designs began to report slamming and
subsequent vibration problems in following seas at
zero and low ship speeds. Yet ferries did not report
these problems. While the hull forms of cruise
ships and ferries are much the same, the operating
profiles for the ships are quite different. Ferries sail
at low speeds inside the harbor, and then come up
to cruising speed and remain there until they reach
their destination. Cruise ships spend large amounts
of time operating at low speeds even in the open
seas. The reported slamming and vibration problem
occurred in very mild sea states when the waves
had a length in the order of the length of the ship.
In the process of investigating the zero speed
following sea stern slamming and vibration
problem, questions began to arise with respect to a
possible hull girder fatigue life problem. Later
issues of ultimate hull girder longitudinal strength
surfaced. These issues brought together the
sciences of hydrodynamics and structural
. .
engineering.
THE: SUBJECT VESSEL
The cruise ship reported on herein is a typical
state-of-the-art design with a twin podded
propulsion system. The particulars of this 1,900
passenger cruise ship are in Table 1. Its hull form is
a variation of a proven design. Figure 1 shows the
aft body plan with pod. Although the stern of the
subject vessel is somewhat flat, it has considerably
more shape aft than other vessels that have reported
stern slamming problems. Note that the body plan
shows a very reasonable buttock rise moving aft,
and section curvature near the skeg. The minimum
deadrise angle exceeds 3 degrees.
Prior to performing the special stern slamming
test program reported here, a conventional
seakeeping model test program was conducted
which included three pressure panels on the bow
flare and one on the flatter portion of stern near the
transom. All of these test results were within
acceptable limits. The purpose of the stern
slamming seakeeping model tests and hull girder
structural finite element investigation was to
determine if sufficient aft shape was present and if
sufficient structural rigidity and strength existed to
avoid the vibration and structural problems present
in those ships that encountered severe slamming.
Length between perp. 233.00 m
Beam 32.20 m
Draft 8.00 m
Displacement 39000 ton
Speed 21 kn
C 14 2.10 m
Table 1 Main dimensions of the cruise vessel.
DWL
6/ ~ 'POD
Figure 1 Aft body of the cruise vessel.
CONSIDERATIONS FOR SLAMMING
MEASUREMENTS
In general, impacts due to slamming contain
energy in high frequency bands (1-20 Hz). At very
low deadrise angles (below 5 de"), the high
frequency contents increase with decreasing
deadrise angle. Also hydroelastic effects might be
important, as reported by Faltinsen [1996] and
Bereznitski and Kaminski [20021.
If the high frequency component of the
slamming pressure is important, it needs to be
accurately measured. This means that the set-up
needs to have a high resonance frequency, at least
twice as high as the frequency content of the
slamming signal.
In addition to this, under water video
recordings of an aft body of a ship slamming in
waves revealed a large number of air bubbles
underneath the stern during the impact. This
indicates that there are strong local effects.
OCR for page 335
The conventional test methodology is to isolate
the aft body of the model from the rest of the
model. The aft body is then connected to the main
portion of the model in such a way as to allow a
system of strain gages to be installed along the
joint. Such a system will have a relatively low
resonance frequency and will hence be insensitive
to high frequency contents in the pressure signal.
The measured force will be a result of the dynamics
of the aft body connection to the hull. Deriving the
impact forces on the stern is not straightforward.
Therefore, this testing method was disregarded.
PROPOSED METHODOLOGY
Both aspects mentioned, the high frequency
content of the input and the local effects, made us
decide to use a large array of pressure sensors to
define the impact. The array used is illustrated in
Figure 2. This set-up allows direct measurement of
pressures including the high frequency content in
the signal, without the dynamics of a strain cause
system.
It is important in this set-up that all
components in the model are as rigid as possible.
This essentially rules out measuring the
longitudinal stresses midships by means of a strain
gauge system, just as a segmented stern model
measuring the total impact force was not
considered acceptable, as discussed earlier.
However, the whipping component on the
vertical bending moment is one of the important
consequences of aft body slamming, so a way to
measure it must be developed. The approach taken
was, to solve the problem in two steps:
1. Measure the impact loads (pressures) on a very
rigid model.
2. Measure the whipping loads on a model cut in
two while the frequency of the 2-node
deformation is scaled from ship to model.
The pressures at the stern were measured for
both the flexible and rigid model. The assumption
was that the total pressure would be the sum of the
impact pressure and the pressure due to the
deformation. If this was the case, based on the
results of the model with different natural
frequencies, it would mean that the hydroelastic
effects could be ignored; at least for practical
purposes.
The objective of step 2 was to demonstrate that
the physics of the problem were understood by
being able to calculate the response of the
segmented model. Using the measured pressures
on the stern of the model allows the calculation of
the impact forces on the stern on the model. The
impact forces are then applied to a mathematical
whipping model of the tow tank model in the water.
The validation then consists of the comparison of
the calculated whipping moment to the measured
values. Positive validation would yield a reliable
tool, when used in conjunction with a detailed
structural model of the ship, for determining the
full scale whinnying resr,on.ce of the shin
Figure 2 Aft body of cruise vessel with podded
propulsion and instrumented with 33
pressure gauges.
MODEL AND INSTRUMENTATION
For the model tests reported here, a wooden
model with a geometric scale ratio of 1:49,
equipped with bilge keels, active stabilizer fins and
podded propulsors was built.
The model was built in two segments with the
cut at Station 10 (midships). The connection was
made via a steel plate that was instrumented to
measure the vertical bending moment (VBM). This
steel plate is illustrated in Figure 3. The gap
between the two halves of the hull was closed with
a thin rubber membrane to make it watertight. The
weight and inertia of both segments were calibrated
separately. A rigid model could be obtained again
by fitting of longitudinal beams along the gunwales
of the model. Accelerometers were fitted on the
fore and aft end of each hull segment so that also
the deformation of the vessel could be measured.
The stern of the vessel was instrumented with
33 pressure pick-ups as illustrated in the photo in
Figure 3 and the drawing in Figure 4. The sampling
rate of the measurements was 286 Hz (full scale).
Anti-aliasing filtering was applied and the
OCR for page 336
measured signals were run through an LP filter
with a cut-off frequency of 95 Hz.
: Regular waves, Arnpl=2~5 m, T=8.0 s, Heading=O:deg
TEST NO 337001
ANALYSIS OF THE IMPACT PRESSURE °
When a slam occurs during the tests in waves,
short duration impulses are measured on each of
the pressure gauges. Figure 5 shows such a
recording. This figure shows that there are
important differences in the peak of the impact
pressure (from 200 to 1000 kPa) and in the timing
of the peak as a function of the different locations
of the gauges. Also, the shape of the pressure
pulses as a function of time can be quite different
from the various sensors.
SLAM 19 200 ~
kPa |
O
CLAM 20 200 ~
kPa o 1 __
200 - l
SLApMe21 1 00 ~ _ _ _
!500
SUM 22 :
kPa
O-
200 -
SLAM 23 .~00
Jew _
.1~,
a:
400 -
SLAM 25 200
. . . .
1.5 1.6 1.7 1.8
SECONDS
Figure 5 Recordings of pressure gauges 18-25
during a slam.
Figure 3 Instrumented steel plate to measure
vertical bending moment.
STARBOARD
15 ~
O O
1 9
0 1°
110
o
3 120
_ _ _~
0 13
i 10
0 1°
O O
~ 17 25
NO
z7o
190 280
20O
Z9O
Z1~
-I=
22 O
230
40 310
32O
33O
STAT. 0 STAT. 1
CENTRELINE
_ PORT-SIDE
Figure 4 Location of the 33 pressure pick-ups
in the stern of the model.
There appears to be a high-pressure ridge that
travels with a rather high velocity over the aft body
of the vessel. The time difference of the pressure
peak passing the different gauges was used to
derive the velocity of the pressure ridge. Figure 6
shows the analysis of a slamming event; this event
was selected from tests in a sea state characterized
by a JONSWAP spectrum, Hs = 4.0 m, Tp = 8.0 s
and a peakedness parameter ? = 3.3. The figure
shows isobars at different time steps, the isobar
shown is the 100 kPa level of the pressure peak
during the rise of the pressure. Figure 6 shows that
the impact started on starboard side, close to the aft
end of the skeg, 2 meters forward of the aft
perpendicular (APP). The distance between the
isobars is used to derive the velocity of the pressure
ridge; these velocities are indicated on the figure.
The initial velocity of the pressure ridge is 13-14
m/s forward and to the starboard side and 21
m/saft. The velocity aft increases to 30 m/s after
0.3 s; side ways it decreases to 6 m/s. The high
pressure ridge covers the full length of the aft body
in 0.5 s after the initial impact; then it expands
mainly sideways, at a velocity of 15-20 m/s to
OCR for page 337
starboard and about 45 m/s to port. The duration of
this total impact is a bout 1 s.
Figure 7 shows a pressure map of the second
stage of this impact, 0.55 s after the initial impact,
when the pressure ridge moves sideways to both
port and starboard. The peak pressure is now on the
order of 100 kPa.
Figure 8, shows the second impact occurring at
two locations simultaneously; one on the center
line about 2 meters aft of the skeg and the second
close to the center line on the starboard side. These
two spots join to one large area only 0.15 s after the
impact; which covers the full length of the aft body.
The high pressure area further expands to the sides
at a speed of 20 - 40 m/s. The total impact lasts
about 0.5 s.
:E s
c'
~ 0
~ ~ . = ..... b~ ., ~
A: so—I ~~ °~
~ 1- o.6 ~ O'6 1"
-10~
-
Analysis of a large number of impacts in
different wave conditions shows that most impacts
have a duration in the range of 0.4 to 0.75 s. This is
illustrated in Figure 9
The impact is shown to be a moving pressure
front with peak pressures of about 300 to 500 kPa
in the initial stage of the impact, reducing to 100 -
200 kPa in the second stage. The width of the high
pressure ridge is about 2.0- 3.0 m. The high
pressure area grows very quickly (velocity 30 to 40
m/s) length-wise until it covers the full length of
the stern. After this it travels sideways at a velocity
of 20 - 30 mls. This makes the total duration of the
impact in the order of 0.5 s for the subject vessel.
.
0 t~ ,°~ a,—~ ~
-
5
O
o
~ IC ~~- ~ ~ e~/~wf~x,~=,~~
X-location w.r.t. APP[m1
Figure 6 Contour plot of moving pressure fields
over the aft-body. The plot gives the 100 kPa isobar
lines at different times and the velocities of the
pressure peak.
in
Pressure
[kPa]
200
1 190
180
170
1~
1~
140
130
~ 1~
~ 110
1 100
90
80
70
60
50
40
30
20
10
J o
' 1 1 1 1 1 , , , ~ 1 , 1 1 1 1
0 10 20
x [m]
Figure 7 Pressure map of the impact shown in
Figure 6. Time is 0.55 s. after the initial
impact.
X-location w.r.t. APPlmJ
Figure 8 Contour plot of moving pressure field
over the aft body after the second
impact showing that the initial impact
takes place at two locations.
Hi,
- .?
0 0.5 1 1.5
SLAM DURATION [s]
Figure 9 Statistics of the duration of the impact
in a 4.0 meter Sea State.
OCR for page 338
PRESSURE INTEGRATION
The next step is to derive the impact force on
the aft body as a function of time. This requires a
pressure integration in space. The width of the
pressure ridge requires a density of pressure gauges
which is far greater than practically possible. With
the current experimental set-up it is possible that
the high pressure ridge is at some instant 'lost'
because it is in between the pressure pick-ups.
Therefore, a proper spatial integration of the
pressure signals requires a careful selection of the
integration method.
A simple and robust pressure integration
technique, denoted (S), was tested first. The
method makes use of a so-called Delaunay
triangulation as illustrated in Figure 10. This
triangulation is unique; no other data points fall
within the circle drawn through the corner points of
each triangle. The triangles are used in the pressure
integration.
The measured time trace of the pressure in a
point is multiplied with one third of the area of all
surrounding panels. In formula:
FZSLAM(t) = ~ P(t)
pressure
pick-ups
~ Al /3 (l)
adjacent
panels i
This procedure uses the complete area
enclosed by all the grid points (329.9 mid. A
similar procedure was used to calculate the center
of effort of the impact force as a function of time.
15~
10t
-
.Q
-
.
-10 -
~ =
-15
2 4 6 8 10 12
X location w.r.t. APP [m]
Figure 10 Delaunay triangulation using all
pressure pick-ups in the aft-body of the
model.
1
14 16 18
For example, the pressure measured in Pie,
see Figure 10, is multiplied by i/3 of the area of
triangles 2-4-7-8 and 10.
The proposed integration method is robust and
simple but it requires a density of pressure gauges
which is high relative to the width of the high
pressure ridge. Since this is not the case with the
current set-up, the resulting force, FZS~AM(t), will
be very spiky due to an over-estimation of the force
at the moment the ridge travels over one of the
pressure gauges, and an under-estimation when the
ridge is just in-between the gauges.
To overcome the expected inaccuracy noted in
the simple integration method, a more advanced
pressure integration is applied. This technique is
denoted (C). This method regards the pressure as a
moving pressure ridge over a panel. The shape of
the pressure ridge ptt) is considered constant for
each panel. Since the velocity V of the pressure
ridge can be determined, the time signal at each
point can be transformed from the time domain to
the spatial domain:
pts)=p~t) V
:~\
! ~
| ~~( Activ~nel
I \~\
100
So
\ 3 pressure signals
L AIL I ^^~ a corner point
_-
(2)
- -5
Leo
160
140
5100
~0
60
140
~ _^v
Figure 11 Impression of moving pressure front
over a panel.
OCR for page 339
i
Est mated time-
duration of stem
A:
C 2.!
In
i
Calculated IMPULSE {kNSl:'
7278 kNs serge method ~
7445 kNs advanced my
0.5
nME [S1
~9
1.S
Figure 12 Result of the Simple and Advanced
pressure integration methods for one
slam.
The direction and velocity of the moving
pressure ridge is determined by the pressure-time
signals at the three corner points. The velocity
vector at each corner point is known, but normally
only one of them actually crosses the subject panel.
The pressure signal at this point is used to calculate
the force on the panel in a strip-wise manner for
each time step. Figure 11 illustrates this spatial
integration process over a single panel.
The result of both pressure integrations is
illustrated in Figure 12. The time trace of the signal
is different; the peak of the signal is very different,
but the total impulse appeared to be almost
identical. The impulse ~Fdtis 7278 Ns for the
simplified integration method and 7445 Ns for the
advanced method. The analysis of other slams with
both methods showed that the difference in impulse
between the two methods is limited to 3%.
MATHEMATICAL MODEL TO
CALCULATE THE WHIPPING LOADS
A schematic of the model in the tank is
illustrated in Figure 13. The model is represented
by two masses, connected by a hinge, a spring and
a damper. Both masses are supported by a
hydrostatic spring. This spring cij consists of the
coefficients C33, C35, C53 and CS5 to take into account
heave and pitch restoring as well as the cross
coupling. Hydrodynamic damping is ignored
because of the high frequencies of interest.
The equations of motions of this two mass
system are built using:
a) The pitch motion equation of the forward part
(index 1) around the CG of this part of the
model, and of the aft part (index 2) around its
CG. The distance from the CG of part 2 to the
impact location is Xs~am:
F.. ~ ~ .
IYYO1 + CS3Z1 + CSSD1 =—CH (01—02 ~—BH (01—02
IA~,82 + CS3Z2 + CSSD2 = CH (01—02 ~ + BH (01—02 )
+ FZSlam Xslam
(3)
b) The heave motion equation of the forward and
aft part:
MFz1 + C33z1 + C3sOl = 0 <4'
M Z2 + C33Z2 + C3s82 + FZslam = 0
c)
The fact that the hinge keeps the two parts
connected. The distance from the local CoG to
the hinge is defined by x~ and x2:
Z2 - X202 = Z1 - Xl81 (5)
Equation (5) is used in equation (4) after which the
two equations of (3) are summed and written as 1
single equation. Similarly equation (4) can be
substituted in equation (3) after which the system
can be written as a differential equation:
M Yt ~ =F(y,t) (6)
in which, M, the 6x6 mass matrix can be derived
from the equations (3) through (6~. The yet) vector
contains the unknown displacements and rotations
and their time derivatives:
ytt) = (Z., Z1,0,, 8,, 82, 02) (7)
The right hand side vector force F(y, t) contains the
restoring forces and the slamming force. The
differential equations are solved using an explicit
Runge-Kutta 4th order scheme.
OCR for page 340
MA
, , , ,1, . . .
cH
BH
MF
~ C hi
, ,,,1,,,,
Figure 13 Dynamic model of the cruise vessel as
it was in the towing tank.
DYNAMIC CALIBRATION
The model system is calibrated by hitting the
model on the extreme aft end and calculating the
response. The response in this respect is either the
moment in the connecting spring or the angular
deformation. The angular deformation is the
deformation mode with the lowest eigen frequency.
The response of this system on a triangular load,
force as a function of time, was calculated, see
Figure 14. The duration of the load was varied
while the impulse, .iFdt, was kept constant for the
different cases.
The results of the calculations are shown in
Figure 15. The duration of the impulse is
normalized by the period of the angular
deformation mode of the system. This figure shows
that for very short impulses it is not the peak value
of the force that is important, but the impulse.
When the duration of the impulse is very long, the
response of the system goes asymptotically to the
static case. The response of the system to a constant
force is also shown in Figure 15; it is denoted the
static response.
The model of the cruise vessel was subjected
to an impulsive load when lying at zero speed in
calm water. The results of these tests, the frequency
of the wet eigen mode of the model and the
damping, are presented in Table 2. The low
damping of the flexible model is a value that could
be expected, see Betts et. al. (1977) and Bishop and
Price (1979~. The high damping of the rigid model
is quite surprising. Most likely there was internal
frictional damping in the fixation of the additional
beams as a result of the high accelerations.
:2000
Q
-2000
. 100
z
Y
50 ~
25
-4000 · , ~ . . n
0 2 ~ ~ ~ 10 12 14 16 18 2
TIME [see]
Figure 14 Whipping response of the model on a
triangular impulsive load.
1.4 -
.2 -
u' 1.0-
o
u' 0.8-
E 0.6-
~ 0.4-
._
x
no
v.
0.0 -
0.0
\ 1'
~
\ |Flexible I
~ T
Dynamic solution
Model
Static solution
.
-
_
0.5 1.0
T impulse / T resonance [-]
Figure 15 Response of the schematized model to
an impulse of varying duration at the
extreme aft end.
2.0
As was illustrated in Figure 9, the duration of
the slamming impact is on the order of 0.5 s. This
allows the points for the rigid and flexible model to
be plotted in Figure 15. The figure shows that the
response of the rigid model is not quasi-static, in
fact the rigid model was not rigid enough. The
response of the flexible model will be closer to a
quasi-static response than that of the rigid model.
The objective of comparing the response of the two
models is to compare the response of two models
with varying flexibility.
The response of the flexible model on the
impulse is shown in Figure 16. This figure shows
the pressure measured by gauge P28 and the local
vertical acceleration. High frequency local
vibrations are dominant in the initial stage of the
impact; these are damped in about 2 seconds. After
this time the simplified 2-node deformation mode
is dominant. About 0.5 s after the initial impact, the
pressure P28 is reasonably in phase with the local
OCR for page 341
vertical acceleration (times -8~. This means that the
measured pressure is only due to the added mass
effect. A similar relation between the local vertical
acceleration and the pressure was found for the
'rigid' model.
rigid model
flexible model
frequency damping
[Hz] [-]
1.75 0.049
0.83 0.0069
Table 2. Results of hammer tests model. 20Q
160
120
80
-120
-160
~ 200 0
Figure 16 Response of flexible model on a
hammer impact. The figure shows the
P28 pressure and the local vertical
acceleration.
This result is used in the analysis of the impacts in
regular waves. The measured load is split into
parts, the first due to the actual wave impact and a
second part due to the local deformation. The
pressure due to the actual wave impact is defined:
PIMPACT = PMEASURED PLOCAL DEFORMATION (8)
For P28 the pressure due to the local deformation
Is:
PLOCALDEFORMATION = 8 AZX=P28 (9)
The results of this analysis are shown in Figure 17.
The difference in the peak values is quite low; and
it is reduced by correcting them for the local
deformations as illustrated in the lower of the two
plots. Note that the local vibrations are dominant
over the 2-node bending mode at the time of the
impact. A comparison of the integrated pressure for
the rigid and the flexible model is shown in Figure
18.
From these results it is concluded that the
hydro-structural interaction is low; the pressure due
to the whipping of the model is an order of
magnitude lower than the pressure resulting from
the impact. Hydro-elastic effects are again lower,
so it is concluded that, for practical purposes, they
can be ignored in the problem of aft body
slamming.
P28rl
P28fl
P28rc
P28fg
O
-100 D
tr
2 3 4
Figure 17 Impact of a regular wave on the flexible
and the rigid model.
Top graph: measured pressures at P28.
Bottom graph: pressures corrected for
local deformation.
OCR for page 342
X10
REGULAR WAVES ~—Flexible model |
~ `; m AAAP! IT' OF I Rioid model I
ad
~2
Cal
in
—~ ~ v—~
Figure 18 Slamming force for the flexible and
rigid model. Regular following waves,
amplitude 2.6 m, period 8 sec.
TUNING OF THE WHIPPING MODEL
The mathematical model requires as input the
properties of the model and the stiffness CH and
damping BH of the connecting spring. The value of
CH follows from the natural period of the 2-node
vibration and by assuming uncoupled motions
between heave and pitch, so by neglecting the z-
displacements in equation (3~:
.. . ~
01 + 2KC)n (31 + COn 01 = 0
_ 2BH 2 4CH + C55 A (1 0)
C,3 . IF+A C) =
n yy YE
The damping BH follows from the whipping
moment decay curve, which was obtained by
hitting the model with a hammer. This damping
curve contains both hydrodynamic and structural
damping. The first is normally considered
negligible at this frequency. The full data of the
proposed mathematical model are listed in Table 3.
Figure 19 presents the correlation between the
mathematical model and the decay curve of the
physical model. The initial conditions for starting
the curve are obtained using the measured vertical
bending moment (MY) at that time. The initial part
of the simulation is shown in the detailed graph in
the figure. Obviously, in the decay curve the
vertical slam force (FZ) is zero. The tuned non-
dimensional damping coefficient was ? = 0.0077;
this is a low value in comparison to full scale ships.
Full scale ships have a damping on the order of 2 to
3% of the critical value.
LINE = MEASURED
CIRCLE = CALCU~TED
110 115 120 125 130 135 140 ~ 145 150 155
TIME [see]
Figure 19 Decay curve of the physical model and
the mathematical model for whipping
due to an impulse (hammer blow).
mass [ton]
kyy [m]
LCG [m]
C33 [kN/m]
C35 = C53 [kN]
C55 [MNm]
CH [MNm]
? [-]
.
Aft Forward
.
22237 16636
40.87 43.81
-53.53 44.30
36521 27521
lW575 -7743
22040 38620
.
4247
0.0077
Table 3. Data of whipping mathematical model.
The data is relative to the CG of the
segment. LCG is relative to the cut.
USE OF THE WHIPPING MODEL TO
ANALYZE SLAMMING EVENTS
The objective of the application of the
whipping model to analyze slamming events was to
check if the physical process was understood. If the
slamming force, as derived from the pressure gauge
measurements, is applied to the whipping model,
the response of this model must correspond to the
response of the model in the tank. When that is the
case, the loads can be applied to a Finite Element
(FE) structural model of the ship as built.
A first check was carried out to compare the
response of the model to the simple integration
method (S) and the advanced integration method
(C). This comparison is made in Figure 21. The
calculations start at T = 381 seconds with initial
conditions zero, thus with no displacement,
rotations or whipping moment present in the hull.
OCR for page 343
::L:ong-cre:sted ~Js, Hs 4.:0 my, TO 8.0 s':Head 0 deg. Speed 0 kn
TEST NO 3:1:5001
x10
MY TO:T
klKlm
-
::M~y WHIPPINGS ~
kNm to a| L
.1 .
11
U ,5~
X o
::2- ~ iN
MY WAVE 1 ~~L,~,7 N; ~~—l a! , ~ T J~ \
SECONDS
Figure 20 Time traces of the Vertical Bending Moment midships. Top: the total signal; Middle: the
whipping component; bottom: wave frequent component.
x 10 10
5 , , . , . . .
simple method (S)
complex method (C)
At
.e
CL
i"
,
;:
,
~ ~ .
- . ~
· :
.. Y
.
.
,
.
~ . -. .
~_c'~
.
.
. .
.
:
:
i.,
381 382
; Or
O
_
385 386 387 S88 389
TIME [sl
Figure 21 Calculated whipping response due to a
single slam.
The slamming force shows some spikes when the
simple integration method is used, but the model does
not respond to such short duration, low impulse
spikes. The result of the simulation shows an
identical response of the model in the sense of the
vertical bending moment
RESULTS OF EXPERIMENTS IN WAVES
Tests were carried out in following sea
conditions at zero speed. The wave conditions
consisted of long crested seas, significant wave
height 2.0 and 4.0 meters, irregular waves using the
JONSWAP spectrum with a peakedness parameter
3.3, and a peak period of 8.0 s. The energy spectrum
of the vertical bending moment midships is shown in
Figure 22. The figure clearly shows two peaks, the
one at a frequency of 0.8 rad/s due to the waves and
the peak at 5.2 rad/s due to whipping.
The measured MY (in de 4.0 m Sea State) is
plotted in Figure 20. The total signal (top time trace)
is split into a whipping component (using a high-pass
filter with cut-off frequency 3 rad/s) and in a wave
component. Both components are also shown in
Figure 20. The vertical bending moment due to
whipping is clearly dominant in this condition. The
results of the whipping model will be compared to
the whipping component of the vertical bending
moment midships; the comparison will mainly be
made for the 4.0 meter Sea State as specified above.
Earlier discussion showed the whipping response
of the vessel due to a hammer impact. The validation
was made using zero initial conditions in the
calculations. In waves the situation is different;
OCR for page 344
Figure 23 shows that whipping due to previous
slamming events still occurs when the vessel is hit by
the next wave.
500.
400 -
.. in
E
it
: ~
~ ?°°-
.~¢
105
0:_
~ 4 6
WAVE FREQUENT U. RADtS
Figure 22 Spectrum of vertical bending moment °
showing distinct wave frequency part
and whipping part. JONSWAP spectrum,
Hs = 4.0 m, Tp = 8.0 s, Vs = 0, following
waves.
6~11
Jr,WHPP09G
10Mm
2
L
_ FZ SUb.
_, kNm
a_" .
- ~!1~'
x104
The timing between the existing whipping
deformation and the next slam impact appears to be
crucial for the resulting response. For example, a
relatively low impact force is able to minimize the
whipping moment at t=26 s. A similar event is found
around t=110 seconds. On the other hand, the impacts
at t=34 and t=41 s. increase the whipping moment
more than could be expected due to a 'favorable'
timing.
The whipping load is calculated using the initial
conditions (angles of the fore and aft segment,
angular velocities) from the model tests. The
amplitude of the whipping load is used to obtain the
difference in the angle of the fore and aft segment.
The angular velocities are obtained from two angular
rotations at consecutive time steps. Figure 24 through
Figure 26 present the results of the simulations with
the whipping model compared to the measured
vertical bending moment for three cases. Figure 24
shows how a relatively small impact increases the
whipping moment some 20%. Figure 25 shows how a
slam that occurs out of phase with the whipping
motion reduces the moment by a factor of four. The
calculations predict the trend correctly, but the
phasing of the whipping moment after the impact is
not correct.
Long~crested Js, his 4.0 m, TO 8.0 5, Head O:~deg, Speed O 1m
TEST NO 315001
X~1o,
20 40 60 ~-30: ~ 00
SECONDS
16 17
Figure 23 Application of the whipping model.
The figure shows a strong relation
between the resulting whipping
moment and the timing of the impact.
l
18 19 20 21 22 23
TIME [s]
24
Figure 24 Example of the measured whipping
response in the towing tank compared
to the response of the whipping model
due to the measured impact force.
OCR for page 345
1 , , ,1
215 216 -217 218 219 220 221
TIME [s]
1
222 223 224
Figure 25 Example of a slamming event that
causes a decrease of the whipping
moment.
Blot
I I I fir, r I , ,
382 -383 ~ ~ 384 385 386 387 388 389 390
TIME [s]
Figure 26 Example of a slamming event that
causes an increase of the whipping
moment.
Figure 26 gives an example of how a favorable
tuning increases the whipping moment with a factor
4. The prediction of the whipping model agrees very
well to the actual measured moment.
The general good agreement between the
calculations using the whipping model with the actual
measured vertical bending moment shows that the
whipping phenomenon is 'understood', and that the
response of the model can be reliably predicted by
the whipping model for individual cases.
STATISTICS OF WHIPPING LOADS
The previous section showed that the amplitude
of the whipping moment is very much dependent on
the timing of the slam relative to the existing
whipping motion. When simulations with the
whipping model were carried out, each slamming
event could be predicted (using the measured
slamming force) if the initial conditions just before
the impact were tuned. If this is not done, inevitably
some drift will occur, which will change the tuning of
the slam with the whipping motion and which will
hence have a large effect on the resulting peak
. .
w ~ppmg moment.
We now consider a long record of measured
whipping moments and compare the statistics of the
measured whipping peaks to those from the whipping
model. The initial conditions for the whipping model
are not tuned to match the experiments; therefore
individual slamming events can be very different
between experiment and prediction.
l .OE+OO
1 .OE-01
1 .OE-02
1 .0E-03
1 .OE-04
l l l l ' ' ' ' 1 ' ' ' '
O Measured
C' ~Iculated
_ _
ot+ ~
~ O
0 500 1000 1500 2000
VBM-whipping [MNm]
Figure 27 Extreme values of the whipping part of
the Vertical Bending Moment compared
to the un-tuned results of the whipping
model. Results from a 4.0 meter Sea
State
The measured peaks of the whipping moment in
the 4.0 meter Sea State are plotted in Figure 27 and
they are compared to the peaks calculated with the
whipping model. The vessel was at zero speed in a
following sea condition. This result shows that the
distribution of the peaks is similar, so that the
differences, which develop due to 'drift' in the
whipping model, are inconsequential to the extreme
whipping moment.
Figure 28 shows a similar plot for a 2.0 meter
significant wave height Sea State. The differences
between actual measurements and the whipping
model at low probabilities of exceedance are
important, but the tails of the distributions are again
similar. These results mean that a simulation with the
whipping model as presented in this paper and using
the measured slamming force cannot be used for
fatigue assessments. However, the results of the
OCR for page 346
whipping model can be used for the prediction of
extreme values.
PREDICTING THE WHIPPING LOADS IN
THE ACTUAL SHIP
The road is now paved to predict wave loads for
the actual ship. This paper demonstrates that the 700-
impact load can be constructed from the measured 600-
pressures. Applying this load to a structural model of
the segmented model as it was in the towing tank. It _ 500 -
is possible to reproduce the whipping bending ~ 400
moment at midships. The next step is now to apply ~
the measured impulse to a full 3D structural model of ~ 300 -
the ship and do a time domain simulation to > 200
determine the whipping response and peak stresses.
inn
n 2
1 .OE+OO ~ .
cut
' 1.0E-01
x
_ 1.0E-02-
a
-
Q 1.0E-03-
0
1 .OE-04
I''' ,lll
.—~ _
__
_ ' ' ' ' 1 ' ' ' '
O Measured
~ Calculated
0 200 400 600 800 1000
VBM-whipping [MNm]
were always significantly lower than those in
following wave conditions at zero speed. This again
illustrates how important the whipping loads in
following waves at very low speeds are for the design
of the ship.
| )K w eve |
| O w hipping |
0 1 2 3 4 5
Significant wave height [m]
Figure 29 Vertical bending moment (mean of 1/3
highest peaks) as a function of
significant wave height. Vessel at zero
speed in following wave conditions.
700
600
500
400
-
300
200
Figure 28 Extreme values of the whipping part of
the Vertical Bending Moment
compared to the un-tuned results of the
whipping model. (2.0 meter Sea State).
o
IMPORTANCE OF THE WHIPPING LOADS o
Although it is not the subject of this paper it is
interesting to note the importance of the whipping
loads for this vessel. Tests in different wave heights
indicate a roughly linear increase of the whipping
moment as a function of the wave height as shown in
Figure 29. The whipping loads quickly decrease as a
function of speed as illustrated in Figure 30.
Many tests were carried out; some of these were
devoted to extreme wave conditions. These tests were
carried out in head and bow quartering waves since it
is considered poor seamanship to be in a following
sea condition in an extreme sea state. The whipping
loads in these conditions (due to bow flare impacts)
\
\
|+wave
0 w hipping
\B
Speed [kts]
6
Figure 30 Vertical bending moment (mean of 1/3
highest peaks) as a function of ship
speed. Following waves condition.
CONCLUSIONS
The objective of this paper was to introduce a
method to reliably measure the excitation of a ship
hull due to aft body slamming impacts. The measured
impact force has been derived from a large array of
pressure gauges. Tests with two models with varying
stiffness indicated that the pressure due to the
OCR for page 347
whipping response of the model could be
superimposed on the impact pressure. This means
that there is hydro-structural interaction, but no
important hydro-elastic effects.
The impact force has been used in a simple
structural schematization of the segmented model that
was used in the towing tank. The predictions with this
whipping model compared very well to the measured
vertical bending moment. It is therefore concluded
that the impact force could be applied to a structural
model of the ship to determine the whipping moment
and peak stresses for the full scale vessel.
Further conclusions are:
· A vessel having a modern stern shape lying at
zero or low speeds in following wave conditions
can experience heavy stern slamming. This
slamming starts occurring in mild conditions if
the wave length is in the order of the ship length.
Stern slamming can be considered as an impact
that starts very locally and expands first
lengthwise and then sideways over the stern. The
high pressure ridge passes a single point in the
stern in about 0.05 s; due to the size of the stern
the total impact duration is about 0.5 s.
Instrumenting the aft body of the vessel with a
large array of pressure gauges allows an accurate
measurement of the impulsive force.
The measured impulsive force can be used on a
dynamic structural model of the actual ship to
predict the whipping loads.
ACKNOWLEDGEMENT
The authors acknowledge the permission of Northrop
Grumman to publish the results in this paper.
REFERENCES
Bereznitski A. and Kaminski M.L., 2002, "Practical
implications of hydroelasticity in ship design",
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survey of hull damping", Transactions RINA. Vol.
119, pp. 125-142.
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Haugen E.M, Faltinsen O.M. and Aarsnes J.V., 1999,
"Application of theoretical and experimental studies
of wave impact to wet deck slamming", Proceedings
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Faltinsen O.M., 1996, "Slamming", Colloquium for
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Faltinsen O.M, 1999, "Water entry of a wedge by
hydroelastic orthotropic plate theory". Journal of
Ship Research, Vol. 43, No 2, pp. 180-193.
Faltinsen O.M., 2000, "Hydroelastic slamming",
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Kurimo R., 1998, "Sea trial experience of the first
passenger cruiser with podded propulsion",
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Representative terms from entire chapter:
pressure ridge
